bhakk

“To minimize B''s time, we must maximize A''s time.
Since each machine can work for at most 8 hours, let A''s time = 8 hours.
Since A''s rate = 9000 pencils per hour, the number of pencils produced by A in 8 hours = rt = 9000*8 = 72,000.
Remaining pencils = 100,000 - 72,000, = 28,000.
Since ...”

“]The average salary of 15 people in the shipping department at a certain firm is $20,000
So, (sum of all 15 salaries)/15 = $20,000
Multiply both sides by 15 to get: sum of all 15 salaries = $300,000
The salary of 5 of the employees is $25,000 each and the salary of 4 of the employees is $16,000 ...”

“Target question: Does each angle in the triangle measure less than 90 degrees?
Given: A triangle has side lengths of a, b, and c centimeters
Statement 1: The 3 semicircles whose diameters are the sides of the triangle have areas that are equal to 3 cm², 4 cm², and 6 cm², respectively. ...”

“Take the task of selecting 4 books and break it into stages.
Stage 1: Select two biology books
Since the order in which we select the books does not matter, we can use combinations.
We can select 2 books from 10 books in 10C2 ways (45 ways)
So, we can complete stage 1 in 45 ways
If ...”

“Target question: What was the number of defective batteries in a shipment of 4000 batteries?
Statement 1: In a particular sample of 10 batteries selected from the shipment, 1 was defective.
Don''t make the mistake of assuming that this means 1/10 of ALL batteries are defective. There''s no ...”

“There aren''t many possible outcomes, so let''s start by listing all possibilities.
Let A, B and C represent the 3 boxes, and let a, b, c, be the keys for those boxes (in that same order).
A = $1 prize, B = $100 prize and C = $1000 prize
If the boxes are arranged as A-B-C, then these are ...”

“http://s27.postimg.cc/531tmnqpr/circ_int.jpg
Answer: E
The important takeaway here is that "intersect" does not necessarily mean "pass through"
So, a line that is tangent to a circle (touching the circle but not passing through it) can be said to intersect the circle. To ...”

“First, let''s take a moment to get a good idea of what this strange notation means.
A few examples:
-1 < x/2 ≤ 0
Multiply all sides of the inequality by 2 to get: -2 < x ≤ 0
Check the answer choices....
Only -3/2 falls inside that range.
Answer: B
Cheers,
Brent”

“Hi Aliya,
I have a feeling that your email system is treating the emails as spam. Have you checked your spam/junk folders?
You might to sign up for the emails again, but this time, be sure to add guide@beatthegmat.com as a contact in your e-mail system. This will help ensure that the daily ...”

“This is a Variables in the Answer Choices question (VIAC). This question type can be answered algebraically or using the INPUT-OUTPUT approach.
Mitch has demonstrated the INPUT-OUTPUT approach, so let''s solve the question algebraically.
The average American spends y dollars on food per month, ...”

“Test the THRESHOLD.
In the question stem, the threshold is 40 CHAIRS.
The total cost of 40 chairs = 40*50 = 2000.
Statement 1: The total price of the desks and chairs is 5,000.
It is possible that 40 chairs are purchased (at a total cost of $2000) or that MORE CHAIRS are purchased.
...”

“Monthly food expense except in January:
Let y=$2.
Discount in January:
Let x=100%.
Since $2 is spent each month for 8 of the first 9 months, the total spent during these 8 months = 8*2 = 16.
Since food purchases are 100% lower in January -- implying that they are FREE in January -- the amount ...”

“Target question: What is the value of a?
Given: 81 divided by a results in a remainder of 1
In other words, 81 is 1 greater than some multiple of a
This means that 80 is some multiple of a
Another way to say this is: a is a divisor of 80
The divisors of 80 are: 1, 2, 4, 5, 8, 10, 16, ...”

“This question is a great candidate for the Double Matrix method, whichcan be used for most questions featuring a population in which each member has two characteristics associated with it.
Here, we have a population of motel rooms, and the two characteristics are:
- air conditioning or no air ...”

“Alternate approach:
To determine the ratio of birds to dogs, plug in a TEST VALUE for the total number of pets.
Let b = the number of birds and d = the number of dogs.
Let the test value for the total number of pets = the LCM of 3 and 5 = 15.
Since 1/5 are birds, b = 1/5 * 15 = 3.
Since 1/3 ...”

“Target question: How many of the pets are dogs?
Given: 1/3 of pets are dogs & 1/5 of pets are birds
Let T = TOTAL number of pets.
So, T/3 = number of dogs
And T/5 = number of birds
Statement 1: There are 30 birds at the pet shop
So, T/5 = 30
This means that T = 150, which ...”

“It turns out that the cost per apple is irrelevant. Here''s why:
The average (arithmetic mean) price of the 10 pieces of fruit is 56 cents
So, (total value of all 10 pieces of fruit)/10 = 56 cents
This means, total value of all 10 pieces of fruit = 560 cents
How many oranges must Mary put ...”

“Target question: How old is Carmen now?
Given: 3 years ago Jennifer was 22 years old
So, Jennifer''s PRESENT age is 25
Statement 1: 10 years ago, Carmen was twice as old as Jennifer.
If Jennifer''s PRESENT age is 25, then Jennifer''s age 10 YEARS AGO was 15
This means Carmen''s age 10 ...”

“No takers?
Let''s say there are 100 apples in the barrel.
45% of the apples weigh MORE THAN 81 grams
This means 55% of the apples weigh LESS THAN OR EQUAL to 81 grams
In other words, 55 apples weigh LESS THAN OR EQUAL to 81 grams
81% of the apples weigh LESS THAN 145 grams
This means 19% ...”

“Statement 1:
If d=72, then t = d/r = 72/72 = 1 hour.
If d=144, then t = 144/72 = 2 hours.
Since the time can be different values, INSUFFICIENT.
Statement 2:
Time and rate have a RECIPROCAL RELATIONSHIP:
If the time is cut in HALF, then the rate DOUBLES.
If the time DOUBLES, then the ...”

“ Given: A dental licensure exam requires a 75% minimum score in order to pass each section
Target question: Did Jennifer pass the 30-question third section?
75% of 30 = 22.5
So, in order to get at least 75% in the 3rd section, Jennifer must correctly answer 23 or more questions in the 3rd ...”

“Target question: Does (a – 2)(b + 4) = 8?
This is a good candidate for rephrasing the target question.
When I SCAN the statements, I see that statement 1 has the term ab. This suggests to me that we might benefit from EXPANDING the expression, since that will result in an ab term.
Take: (a ...”

“We want P(select at least one 6)
When it comes to probability questions involving "at least," it''s best to try using the complement.
That is, P(Event A happening) = 1 - P(Event A not happening)
So, here we get: P(getting at least one 6) = 1 - P(not getting at least one 6)
What does ...”

“GIVEN: x and y are consecutive integers such that x < y
So, if x is ODD, then y is EVEN
One option here is to replace the x''s and y''s with ODD and EVEN and apply the rules for ODDs and EVENs.
Or we can just replace x and y with odd and even numbers. Let''s do that.
Replace x with 1 ...”

“Let G = the gas in the tank and C = the capacity of the tank.
Statement 1:
G - (1/2)G = (1/3)C
No way to solve for G.
INSUFFICENT.
Statement 2:
G + 0.75 = C
No way to solve for G.
INSUFFICIENT.
Statements combined:
Since we have two variables (G and C) and two distinct linear ...”

“Target question: What percentage of his bills did Stanley pay before their due dates last month?
This is a great candidate for rephrasing the target question.
Aside: We have a free video with tips on rephrasing the target question:
Let P = # of bills PAID before due date
Let T = TOTAL # of ...”

“If the area of the square is 4, then each side has length 2
https://i.imgur.com/Ul16WsZ.png
At this point, we have a special 30-60-90 right triangle. When we compare this blue triangle to the BASE 30-60-90 right triangle . . .
https://i.imgur.com/vrYSacE.png
. . . we see that the blue ...”

“Here, the value in red is the sum of Jake''s weight and TWICE the sister''s weight:
187 + 2(89.5) = 366.
But the prompt gives an integer value for the sum of Jake''s weight and the sister''s ACTUAL weight -- a constraint that forces Jake''s weight to be an even integer.”

“The sum of Jake''s weight and his sister''s weight is an INTEGER VALUE.
The answer choices indicate that Jake''s weight is an INTEGER VALUE.
Implication:
The sister''s weight must also be an integer value.
Thus, the value in red -- 89.5 -- is not a valid option for the sister''s weight.
Given ...”

“Set A = {2, 3, 4, 5}, and set B = {4, 5, 6, 7, 8}. If P = the product of one number chosen from set A and one number chosen from set B, how many DIFFERENT values of P are possible?
A) 14
B) 15
C) 16
D) 18
E) 20
Answer: C
Difficulty level: 600 - 650
Source: www.gmatprepnow.com”

“Let the river''s rate = 1 mph and the boat''s rate = 5 mph.
When the boat travels DOWNRIVER, the boat and the river WORK TOGETHER, so we ADD their rates:
5+1 = 6 mph.
When the boat travels UPRIVER, the boat and the river COMPETE, so we SUBTRACT their rates:
5-1 = 4 mph.
Rate and time have ...”

“ALWAYS KEEP YOUR EYE ON THE ANSWER CHOICES.
If Jake loses 8 pounds, he will weigh twice as much as his sister.
j-8 = 2s
j = 2s + 8 = even + even = even.
Since Jake''s weight must be an EVEN VALUE, the correct answer is E.”

“Here''s a solution that uses one variable.
Let x = Jake''s present weight in pounds
So, x - 8 = Jake''s hypothetical weight IF he were to lose 8 pounds
If Jake loses 8 pounds, he will weigh twice as much as his sister.
In other words, the sister weighs HALF as much as Jake''s hypothetical ...”

“The main concept here is that all 3 children are equally likely to win this game (unless one of them possesses supernatural powers that allow him to know what numbers the other two boys will choose :-))
Also note that, if everything is random, the probability of winning by choosing the number 2 ...”

“In a barrel full of apples, 45% of the apples weigh more than 81 grams, and 81% of the apples weigh less than 145 grams. What percent of the apples in the barrel weigh more than 81 grams but less than 145 grams?
A) 16
B) 18
C) 26
D) 32
E) 36
Answer: C
Difficulty level: 600 - 650
Source: ...”

“This is one of my all-time favorite questions!!
The main concept here is that all 3 children are equally likely to win this game (unless one of them possesses supernatural powers that allow him to know what numbers the other two boys will choose :-))
Also note that, if everything is random, ...”

“Let S = Sara''s age.
Sum of the ages = 2+4+5+8+10+13+S = 42+S.
Statement 1:
Since the average age is equal to an INTEGER -- the fourth largest age -- we get:
Sum of the ages = (number of ages)(average age) = (7)(INTEGER) = multiple of 7.
The blue expression and the red expression each ...”

“From 2000 to 2003, the number of employees at a certain company increased by a factor of 1/4
Let x = the number of employees in 2000
So, the number of employees in 2003 = x + (1/4)x
= (5/4)x
From 2003 to 2006, the number of employees at this company decreased by a factor of 1/3
(5/4)x = the ...”

“Target question: How much money could be saved by purchasing 10 bagels at the sale price rather than at the usual price?
Statement 1: The usual price for a bagel is $0.50 per bagel.
Since there''s no information about the SALE price, there''s no way to calculate the amount saved.
Since we ...”

“We can PLUG IN THE ANSWERS, which represent the value of m.
When the correct answer is plugged in, the median will be equal to the range.
Note:
When the prompt includes the phrase which of the following, the correct answer is likely to be D or E.
E: m=3
Resulting set: 1, 1, 2, 2, m=3, 3, 3, ...”

“Let X=8 and Y=4.
Henry eats X scones in X percent of the time it takes Rachel to eat Y scones.
Since Henry eats X=8 scones in 8% of the time it takes Rachel to eat Y=4 scones -- 10 minutes, according to the prompt -- the time for Henry to eat 8 scones = 8% of 10 minutes = (8/100)(10) = 4/5 ...”

“Let''s use the INPUT-OUTPUT approach.
Let''s assign some values of X and Y so that they satisfy the information in the question.
Let X = 25. This means Henry eats 25 scones in 25 percent of the time it takes Rachel to eat Y scones
Let Y = 100. This means Henry eats 25 scones in 25 percent of ...”

“I''m not crazy about this question.
The official answer suggests that, although 2 books are DUPLICATES, they''re still considered DIFFERENT.
Here''s what I mean:
Let''s let D and d represent the two duplicate books.
Let, A, B and C represent the other three books.
The official answer ...”

“ Given: x and y are integers
Target question: Is x² + x + y an odd integer?
This is a good candidate for rephrasing the target question.
Take x² + x + y and factor the first two terms to get: x(x + 1) + y
Notice that x and x+1 are CONSECUTIVE INTEGERS, which means one value is ODD and ...”

“Consider the following equation:
7x + 5y = 70.
If x and y are nonnegative integers, the following solutions are possible:
x=10, y=0
x=5, y=7
x=0, y=14.
Notice the following:
The value of x changes in increments of 5 (the coefficient for y).
The value of y changes in increments of ...”

“In order to minimize the number of homeowners, we must MAXIMIZE the number of men in the group, since the proportion of male homeowners (20%) is less than the proportion of female homeowners (25%)
So, let''s see what happens if there are 199 men and 1 woman.
If 20% (aka 1/5) of the men are ...”

“x is directly proportional to y implies the following:
x/y = k.
x = ky.
In each case, k is a CONSTANT.
The cost of a square slab is proportional to its thickness and also proportional to the square of its length.
Thus:
c = k * T * L².
What is the cost of a square slab that is 3 meters ...”

“species may be singular or plural.
Passage:
Planting trees primarily from the species that turns out to have the best record of survival will assure downtown Thomasville of an abundant tree population 20 years from now.
Here, the verb attributed to species is turns.
Since turns is singular, ...”

“(number of machines)(number of hours) = number of toys that are produced.
In the prompt above, the blue values are THE SAME.
Implication:
2 machines take h+3 hours to produce the same number of toys as 3 machines in 2 hours:
2(h+3) = 3h
2h + 6 = 3h
6 = h.
Since 3 machines produce 360 toys ...”

“...the probability that it
So, 1/4 of the 120 marbles are white.
In other words, there are 30 white marbles.
...the probability that it
So, 1/3 of the 120 marbles are green .
In other words, there are 40 green marbles.
Altogether, there are 70 marbles that are EITHER white or green.
...”

“Total # of combinations = (# of ways to select 2 chairs)(# of ways to select 2 tables)
So, 150 = (# of ways to select 2 chairs)(# of ways to select 2 tables)
# of ways to select 2 chairs
5 tables, choose 2 of them.
Since the order of the selected chairs does not matter, we can use ...”

“Let x = 5 seconds for the first lap, implying 10 seconds for the second lap, 3 seconds for the third lap, and a total of 18 seconds for all 3 laps.
(18 seconds)/(3 laps) = (6 seconds)/(1 lap) = 1/10 minute per lap.
The correct answer must yield a value of 1/10 when x=5.
Only D works:
(x+1)/60 ...”

“The information in red is irrelevant.
Only the following facts matter:
Ross travels onward at a rate of 1 mile every 12 minutes.
He travels home for twice the onward distance at half the onward speed.
Onward speed = 1 mile per 12 minutes = 5 miles per 60 minutes = 5 mph.
Let the onward ...”

“The original total cost of the 10 pieces of fruit = 10*56 = 560.
According to the answers, after 1, 2, 3, 4, or 5 oranges are removed -- so that 9, 8, 7, 6, or 5 pieces of fruit remain -- the average cost per piece decreases to 52.
Since the original total cost is a multiple of 10, and the price ...”

“It turns out that the cost per apple is irrelevant. Here''s why:
The average (arithmetic mean) price of the 10 pieces of fruit is 56 cents
So, (total value of all 10 pieces of fruit)/10 = 56 cents
This means, total value of all 10 pieces of fruit = 560 cents
How many oranges must Mary put ...”

“Take the task of arranging the 10 books and break it into stages.
Stage 1: Arrange the 4 books about Abe Lincoln in a row
We can arrange n objects in n! ways.
So, we can arrange the 4 books in 4! ways
IMPORTANT: Now we''ll "glue" the 4 Abe Lincoln books together to form 1 SUPER ...”

“An alternative approach is to test the answer choices
First, however, we can eliminate some answer choices by first testing an easy value: 50%
Is 25 50% greater than 15?
Well, 50% of 15 = 7.5
So, 50% greater than 15 = 15 + 7.5 = 22.5
22.5 is too small (we need to get 25), so we need a ...”

“Let x = the product of the two denominators in the prompt = 3*4 = 12.
Since 1/3 won a medal for swimming, the number who won a medal for swimming = (1/3)(12) = 4.
Since 1/4 of the medal winners for swimming also won a medal for diving, the number who won both a medal for swimming and a medal for ...”

“Let $100 = share price at the beginning of the year
At the end of the first quarter, the share price of a certain mutual fund was 20 percent higher than it was at the beginning of the year.
So, $120 = share price at the end of the first quarter
At the end of the second quarter, the share ...”

“Given two sides of a triangle, the greatest possible area will be achieved if the two sides form a RIGHT ANGLE and serve as the triangle''s base and height.
For the problem above, I posted an illustrated solution here:
...”

“One approach is to assign a NICE NUMBER to the volume of the swimming pool. This number will work well with the two pieces of information (fill pool in 9 hours and fill pool in 5 hours).
So, let''s say that the volume of the pool is 45 gallons.
The water from one outlet, flowing at a constant ...”

“Target question: Is x > y?
This is a good candidate for rephrasing the target question.
Since y is POSITIVE, we can safely take the inequality x > y and divide both sides by y to get: x/y > 1
REPHRASED target question: Is x/y > 1?
Aside: Here’s a video with tips on rephrasing ...”

“When the SAME DISTANCE is traveled at two different speeds, the average speed for the entire trip will be CLOSER TO THE LOWER SPEED than to the higher speed.
The reason:
The portion traveled at the lower speed takes LONGER.
Since more time is spent at the lower speed than at the higher speed, the ...”

“Average speed = (TOTAL distance)/TOTAL travel time)Carlos drove to work at an average speed of 50 miles per hour
Time = distance/speed
So, travel time TO work = d/50 hours
Carlos drove home from work at an average speed of 70 miles per hour
So, travel time FROM work = d/70 hours
So, TOTAL ...”

“He spent 40% (of the $35) on chicken feed.
40% of $35 = $14. So, he spent $14 on chicken feed.
He got a 20% discount on the chicken feed.
In other words, he paid 80% of the original price.
So, let C = original price of chicken feed.
We can write 80% of C = $14
Or we can write 0.8C = $14 ...”

“On Monday, the opening price of a certain stock was $100 per share and its closing price was $110 per share.
On Tuesday the closing price of the stock was 10 percent less than its closing price on Monday,
We can say that the stock LOST 10% of its value.
OR we can say that the stock RETAINED 90% ...”

“Statement 1: The numbers of pens, pencils, and pads that each staff member received were in the ratio 2:3:4, respectively.
Since 2+3+4 = 9, the total number of pens + pencils + pads must be a multiple of 9.
No way to determine the exact number of staff members.
INSUFFICIENT.
Statement 2: ...”

“Jay''s solution is great.
I just wanted to mention that this question hinges largely on the fact that the number of pens, pencils, and pads that each staff member receives MUST HAVE INTEGER VALUES.
So, for example, if statement 1 were different and statement 2 read "The manager ...”

“Another approach is to test the answer choices
A. If the units digit of x is 1, is it the case that units digits of both (x+2)² and (x-2)² is 9?
Well, (x+2)² = (1+2)² = 9. Works!
To test (x-2)², let''s let x = 11. We get: (11-2)² = 9² = 81. NO GOOD
ELIMINATE A
B. If the units digit ...”

“Both of the following are viable:
Vehicles powered by natural gas emit less of certain pollutants than those burning gasoline or diesel fuel.
Vehicles powered by ndatural gas emit less of certain pollutants than do those burning gasoline or diesel fuel.
The inclusion of do is optional.
With or ...”

“What is the value of n-m?
(m,0) and (n,0) are the two points where y=0 and the graph intersects the x-axis.
Thus, m and n are the x-intercepts of the graph.
Since m < n, n-m > 0.
Question rephrased: What is the positive difference between the x-intercepts?
Statement 1: 4b = a²-4
Test ...”

“I''d typically assign a "nice" value to the original price, and then follow the steps that Mitch demonstrated.
Alternatively, we can solve the question algebraically
In 1991, the price of a house was 80% of its original price.
Let x = the original price
So, 0.8x = the 1991 price
...”

“percent may serve as a NOUN.
Of the students in the class, eighty percent passed the test.
Here, eighty percent is a NOUN serving as the subject of passed.
twice is a MODIFIER.
It cannot serve as the subject of a sentence.
Of the students in the class, twice passed the test.
The sentence ...”

“Another approach is to use algebra.
After her birthday, Fiona had D dollars in gift money. She spent 3/5 of this on an electric skateboard.
If Fiona SPENT 3/5 of D dollars, then 2/5 of D dollars REMAINED
In other words, the amount remaining at the end of the day = 2D/5 dollars
The next day, ...”

“One option is to WORK BACKWARDS
The next day, she spent 1/3 of what was left on a movie passcard, and then finally put the last $80 in the bank.
If she spent 1/3 of her money on a movie passcard, then 2/3 of her money remained (which she put in the bank).
This means 2/3 of her money = $80
We ...”

“ Given: A retailer sells only radios and clocks, and she currently has 44 total items in inventory.
Target question: How many radios are there in the inventory
Statement 1: The retailer has more than 28 radios in inventory
There are several scenarios that satisfy statement 1. Here are ...”

“The error is highlighted above in blue.
if a>0 we will have the answer as YES is ALMOST true.
The only time it''s not true is when a = 1
If a = 1, then the answer is NO.
We can verify this using either the original inequality or the rephrased inequality.
Take "Is a + 1/a > ...”

“Consider the following equation:
5x + 7y = 70.
If x and y are nonnegative integers, the following solutions are possible:
x=0, y=10
x=7, y=5
x=14, y=0.
Notice the following:
The value of x changes in increments of 7 (the coefficient for y).
The value of y changes in increments of 5 (the ...”

“The ratio of milk and water in a mixture is 2:1,
Let milk = 2 liters and water = 1 liter.
Some quantity of water is added to the mixture to make the ratio of milk and water in the new mixture 1:2.
Since the resulting amount of water must be double the original 2 liters of milk, the resulting ...”

“Let the 7 cars be A, B, C, D, P, Q and S.
Since P and Q must occupy adjacent positions, consider PQ a single element in the arrangement.
The number of ways to arrange the 6 elements A, B, C, D, PQ and S = 6! = 720.
In 1/2 of these arrangements, S will be to the LEFT of PQ.
In the ...”

“This question illustrates a common trap on the GMAT.
As Jay has shown, with statement 2, we''re able to write the equation 23x + 11y = 170 , and in high school we learned that, if we''re given 1 equation with 2 variables, we cannot find the value of either variable.
However, if we restrict the ...”

“This Date Sufficiency question is almost identical to the following Problem Solving question, which requires us to find the actual area: http://www.beatthegmat.com/plz-explain-t270003.html
Cheers,
Brent”

“Let''s draw a rectangle around the triangle (as shown below) and then subtract from the rectangle''s area (28) the areas of the 3 right triangles that surround the triangle in question.
We get the following:
http://s13.postimage.org/a47qx0j8j/triangle_in_coordinate_plane_2.jpg
So, the area ...”

“Target question: What is the perimeter of quadrilateral Q?
If we recognize that each statement alone is not sufficient, we can jump straight to......
Statements 1 and 2 combined
There are several quadrilaterals satisfy BOTH statements. Here are two:
Case a: Q is a rectangle with width 1 ...”

“Let $100 be the original price
The price of a consumer good increased by p% during 2012
p% = p/100, so a p% INCREASE is the same a multiplying the original price by 1 + p/100
So, the new price = ($100)(1 + p/100)
The price then decreased by 12% during 2013
A 12% DECREASE is the same a ...”

“The number of students who attend a school could be divided among 10, 12, or 16 buses, such that each bus transports an equal number of students.
This tells us that the TOTAL number of students is a multiple of 10, 12 and 16
What is the minimum number of students that could attend the school?
...”

“Target question: What is the value of y - x?
Given: Set K consists of a finite number of consecutive odd integers
Statement 1: The average (arithmetic mean) of set K is –36
There''s a nice rule that says, "In a set where the numbers are equally spaced, the mean will equal the ...”

“Since the answer choices are so spread out, we can be quite aggressive with out estimation.
First, 10⁸ - 10² can be rounded to 10⁸
Is this reasonable? Well, 10⁸ equals 100,000,000, so subtracting 10² (aka 100) from 100,000,000 still leaves us with approximately 100,000,000
...”

“Let:
W = the total number of white marbles.
E = the total number of even marbles.
WE = the total number of white even marbles.
Statement 1:
Thus, WE = 0.
No way to determine the value of W or E.
INSUFFICIENT.
Statement 2:
Implication:
W-E = (0.2)(25) = 5.
Case 1: W=10, E=5, WE=0
...”

“Statement 2: Brittany always leaves the hotel at exactly 5.30pm.
To see the situation more clearly, plug in a value for the distance between the hotel and the office.
Let the distance = 6 miles.
Since Allison takes 1 hour -- from 5 to 6pm -- to travel the entire distance, Alison''s rate = d/t = ...”

“Target question: Did the profit exceed $4 million on sales of 380,000 units?
Given: The profit from the sale of a certain appliance increases, though not proportionally, with the number of units sold.
IMPORTANT: This means that we CANNOT make any "good" assumptions about the ...”

“It is possible for back-to-back modifiers joined solely by a comma to have the same referent.
Three examples:
http://www.beatthegmat.com/the-hognose-snake-puts-on-an-impressive-bluff-t93558.html
http://www.beatthegmat.com/parallelism-issue-t96011.html
...”

“The usage of the term arrangements is misleading. The prompt seems intended to ask the following:
The first player selected can be paired with any of the 5 other players.
Thus, the number of options for the first player selected = 5.
At this point, 4 players remain.
The next player ...”

“I thought I''d point out that Jay''s approach is also known as the Double Matrix Method.
This technique can be used for most questions featuring a population in which each member has two characteristics associated with it.
Here, we have a population of construction worker, and the two ...”

“Target question: How many of the numbers are less than 35?
Given: The average of four numbers is 35
All this tells us is that the SUM of the 4 numbers is 140 (since 140/4 = 35)
Given the limited amount of information, it''s likely that the correct answer is C or E
Statement 1: None of ...”

“The determine the percent decrease, we need to know the RATIO between the remaining amount of sugar after cookie-making and the original amount of sugar.
Statement 1:
Case 1: Remaining sugar after cookie-making = 10 pounds
Since the remaimning amount of sugar would need to increase by 30% to ...”

“Statement 1: The trip from Townsend to Smallville took 50% longer than the trip from Smallville to Townsend.
Case 1: Distance between Townsend and Smallville = 80 miles
Time to travel from Townsend to Smallville at a speed of 40 mph = 80/40 = 2 hours.
Time to travel from Smallville to Townsend = ...”

“Let x = the number of 100-point labs and y = the number of 85-point labs.
Question stem, rephrased:
What is the value of x?
Statement 1:
100x + 85y = 1140
20x + 17y = 228
The value in blue implies the following:
Since 20x will have a units digit of 0, 17y must have a UNITS DIGIT OF 8.
...”

“Alternate approach:
Since an absolute value cannot be negative -- implying that the right side of the equation must be NONNEGATIVE -- we know that x≥0.
If x²-6=x, we get:
x²-x-6=0
(x-3)(x+2) = 0
Only the factor in green implies a nonnegative option for x:
x-3=0
x=3.
The correct ...”

“ Given: m and n are positive integers such that m > n
Target question: What is the remainder when m² – n² is divided by 21?
Statement 1: The remainder when (m + n) is divided by 21 is 1
There are several values of m and n that satisfy statement 1. Here are two:
Case a: m = 12 and ...”

“Target question: What is the value of (ax + ay)/(x + y)?
This is a good candidate for rephrasing the target question.
Take the expression: (ax + ay)/(x + y)
Factor the numerator to get: a(x + y)/(x + y)
Simplify the fraction to get: a
Since (ax + ay)/(x + y) = a, we can rephrase the target ...”

“Target question: Can n be expressed as the difference of 2 prime numbers?
Statement 1: (n-17)(n-21) = 0
This tells us that EITHER n = 17 OR n = 21
IMPORTANT: Upon seeing that n can equal one of two values, some students will incorrectly conclude that statement 1 is not sufficient.
However, ...”

“The usage of number implies that we should consider NON-INTEGER values.
Both statements are satisfied by z=18.1 and z=19.1
Since the units digit of z can be different values, the two statements combinecd are INSUFFICIENT.
The correct answer is E.”

“In order to minimize the number of homeowners, we must MAXIMIZE the number of men in the group, since the proportion of male homeowners (20%) is less than the proportion of female homeowners (25%)
So, let''s see what happens if there are 199 men and 1 woman.
If 20% (aka 1/5) of the men are ...”

“Target question: What is the value of the integer n?
Statement 1: n is a prime factor of 21
21 = (3)(7)
So, EITHER n = 3 OR n = 7
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: n is a factor of 49
49 has three positive factors: 1, ...”

“Any BAG taken on the subway IS subject to search by the police.
Any BAGS taken on the subway ARE subject to search by the police.
In each of these sentences, any is a MODIFIER describing the subject in blue.
In the first sentence, the verb (is) is SINGULAR to match the singular subject in blue.
...”

“Statement 1:
No information about Charlie or Daryl.
INSUFFICIENT.
Statement 2:
For every 2 home run hits made by Charlie and Daryl combined, Brady made 4 home run hits, implying that Brady made 4 of every 6 home run hits:
4/6 = 2/3.
SUFFICIENT.
The correct answer is B.”

“IMPORTANT RULE: If two sides of a triangle have lengths A and B, then . . .
DIFFERENCE between A and B < length of third side < SUM of A and B
Let''s focus on this part: length of third side < SUM of A and B
We can also say that the length of LONGEST side must be less than the SUM of ...”

“ Given: Brady, Charlie, and Daryl play on the same baseball team.
Target question: The number of home runs hit by Brady in a particular season is what fraction of the total home runs hit by the three players in that season?
This is a good candidate for rephrasing the target question.
Let ...”

“Target question: What is the 1st term in sequence S ?
Statement 1: The 3rd term in S is 2
Since we''re not given any information about the NATURE of the sequence, there''s no way to answer the target question with certainty.
So, statement 1 is NOT SUFFICIENT
Statement 2: The 2nd term in ...”

“Downhill speed - uphill speed = (x+10) - (x-8) = 18.
Implication:
The downhill speed is 18 mps greater than the uphill speed.
We can PLUG IN THE ANSWERS, which represent the downhill speed.
Since the downhill speed is 18 mps greater than the uphill speed, the correct answer must be GREATER ...”

“Target question: Marcia''s bucket can hold a maximum of how many liters of water?
In other words, what is the TOTAL CAPACITY of Marcia''s bucket?
Statement 1: The bucket currently contains 9 liters of water.
Not enough info to determine the TOTAL CAPACITY of Marcia''s bucket.
Since we ...”

“Let''s start from the bottom and work our way up:
We''ll start with the 16th board counting from the bottom of the stack
Right BELOW that 16th board, we have a board that we''ll call Larry.
So, Larry is the 15th board counting from the bottom of the stack.
So, starting from the bottom and ...”

“I thought I''d point out that Manasa3190''s approach is known as the Double Matrix Method. This technique can be used for most questions featuring a population in which each member has two characteristics associated with it.
Here, we have a population of people, and the two characteristics are:
...”

“IMPORTANT: Let''s use HOURS throughout our solution, and then convert hours to minutes at the end.
When car Y had traveled for z HOURS, car Y had traveled 23 miles more that car X had from the time that car X left Town T.
z = car Y''s travel time in HOURS
So, z + 3/10 = car X''s travel time in ...”

“Statement 1:
To MINIMIZE the number of coins so that a sum of 170 cents is yielded by at most 8 coins, we must MAXIMIZE the number of quarters.
Since 7*25 = 175 cents, the number of quarters must be less than 7.
170 cents will be yielded by 6 quarters (totaling 150 cents) and 2 dimes (totaling 20 ...”

“Let the tank = 36 gallons.
Since the first pipe takes 3 hours to fill 1/2 the tank -- and (1/2)(36) = 18 gallons -- the rate for the first pipe = w/t =18/3 = 6 gallons per hour.
Since the second pipe takes 6 hours to fill 2/3 of the tank -- and (2/3)(36) = 24 gallons -- the rate for the second ...”

“Strategy:
Try to show that four of the answer choices do NOT have to be positive.
Case 1: x=1 and y=1
In this case, A, B and D are not positive.
Eliminate A, B and D.
Case 2: x=1 and y=2
In this case, C is not positive.
Eliminate C.
The correct answer is E.”

“Total amount of feed = (number of chickens)(number of days).
The number of chickens is INVERSELY PROPORTIONAL to the number of days.
If the number of chickens DOUBLES, then the feed will last for 1/2 the number of days.
If the number of chickens TRIPLES, then the feed will last for 1/3 the ...”

“Take the task of shipping the 4 boxes and break it into stages.
Let''s call the 4 boxes A, B, C and D
Stage 1: Select a truck to ship box A in
There are 4 trucks from which to choose, so we can complete stage 1 in 4 ways
Stage 2: Select a truck to ship box B in
There are 4 trucks ...”

“ Given: A quorum of at least half the 20 directors must pass the resolution by at least a two-thirds majority
Target question: At a meeting of the board of directors, did resolution X pass or fail?
Statement 1: Ten directors voted for the resolution.
KEY: We don''t know how many ...”

“ Given: Every student at the Performing Arts Academy must take at least one of the two drama courses offered, Classical Theater or Improvisation. 15% of the students who take Classical Theater also take Improvisation
Let''s use the Every student at the Performing Arts Academy must take at least ...”

“ Given: At a certain department store present-wrapping counter, each clerk will wrap no fewer than twenty and no more than thirty presents per hour.
Target question: If seventy people are standing in line, will all of their presents be wrapped after one hour?
Statement 1: Each person in ...”

“Let y = the height of the shorter tree
The taller tree is x times the height of the shorter tree. Ho
So, yx = the height of the taller tree
The two trees have a combined height of 60 feet
We can write: y + yx = 60
How tall is the shorter tree, in terms of x?
We must solve the equation ...”

“Let''s pick some nice numbers that adhere to the given information.
...the constant rate of one pump was 1.5 times the constant rate of the other
So, the fast pump has a pumping rate that''s 1.5 faster then the slow pump.
So, let''s say the SLOW pump pumps at 2 gallons per hour
This means ...”

“ Given: There are two types of rolls on a counter, plain rolls and seeded rolls.
Let P = # of plain rolls
Let S = # of seeded rolls
Target question: What is the value of P + S?
Statement 1: The ratio of the number of seeded rolls on the counter to the number of plain rolls on the ...”

“Take the task of feeding the 5 friends and break it into stages.
We’ll begin with the most restrictive stage(s).
Stage 1: Select a pizza for Phoebe
Since Phoebe will only eat Margherita pizza, there''s only 1 pizza we can serve her.
So, we can complete stage 1 in 1 way
Stage 2: ...”

“Let J = the number of hours that John works.
Since John assembles 3 chairs per hour, the number of chairs assembled by John = 3J.
Let L = the number of hours that Larry works.
Since Larry assembles 4 chairs per hour, the number of chairs assembled by Larry = 4L.
Since John and Larry work a total ...”

“Number of options for Phoebe = 1. (Must be Margherita.)
Number of options for Ross = 3. (Of the 4 remaining pizzas, any but Hawaiian.)
Number of options for Chandler = 3. (Any of the 3 remaining pizzas.)
Number of options for Joey = 2. (Either of the 2 remaining pizzas.)
Number of options for ...”

“Target question: Does this month have 30 or 31 days?
IMPORTANT: Notice that the question does not ask "How many days are in this month?" If that were the case, then we''d have to consider February. Instead, the question is ruling out February by basically telling us that the month in ...”

“One approach is to count all possible arrangements (that may or may not break the given rule) and then subtract all outcomes the break the rule.
Here''s another approach.
Take the task of arranging the 5 letters and break it into stages.
Stage 1: Arrange the 3 CONSONANTS (D, G and T) in a ...”

“Target question: What is the 105th term of S?
Given: Sequence S has 120 terms
Statement 1: The first term of S is −8.
We have no information about the nature of the sequence.
So, knowing the value of term 1 won''t help is determine the value of term 105
Since we cannot answer the ...”

“A fast and easy approach is to assign a "nice" value to the monthly rent of a ONE-bedroom apartment.
Let''s say the rent on a ONE-bedroom apartment is $100 per month
A two-bedroom apartment costs 50% more to rent than a one-bedroom apartment
50% of $100 = $50
So, the rent on a ...”

“Target question: What is the value of x?
Given: If the remainder when positive integer x is divided by 7 is 4
When it comes to remainders, we have a nice rule that says:
If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For ...”

“Target question: How much money could be saved by purchasing 10 bagels at the sale price rather than at the usual price?
Statement 1: The usual price for a bagel is $0.50 per bagel.
Since there''s no information about the SALE price, there''s no way to calculate the amount saved.
Since we ...”

“One woman and one man can build a wall together in two hours, but the woman would need the help of two girls in order to complete the same job in the same amount of time.
Since replacing 1 man with 2 girls does not change the amount of time required to complete the job, 1 man is the equivalent of 2 ...”

“One approach is to determine the size of the job
One woman and one man can build a wall together in two hours, but the woman would need the help of two girls in order to complete the same job in the same amount of time.
The part in blue tells us that 2 girls have the same output as 1 man.
...”

“Target question: What is the y-intercept of line L?
IMPORTANT: When I scan the two statements, I see that they are strongly related.
If the x-intercept is -1/3 (as per statement 2), the slope of the line is GUARANTEED to be 3 times its y-intercept (as per statement 1).
Here''s why...
...”

“The price of the radio was the 15th highest price
So, there were 14 items that were more expensive than the radio.
In other words, if we listed all of the prices in ascending order, we''d get something like:
........... radio, X, X, X, X, X, X, X, X, X, X, X, X, X, X
The price of the radio ...”

“Take the task of creating 4-digit numbers and break it into stages.
We’ll begin with the most restrictive stage.
Stage 1: Select the first digit (to go in the thousands place)
Since the 4-digit number must be GREATER THAN 3000, the first digit must be greater than or equal to 3
This means ...”

“ Given: Automobile A is traveling at two-thirds the speed that Automobile B is traveling.
Let A = Car A''s speed
Let B = Car B''s speed
So, we can write: A = (2/3)B
Target question: What is the value of A?
Statement 1: If both automobiles increased their speed by 10 miles per hour, ...”

“This is a MIXTURE problem.
Two speeds (3 mph and 8 mph) are combined to form a mixture with an average speed of 4 mph.
To determine how much time must be spent at each speed, we can use ALLIGATION.
Let U = the uphill speed and D = the downhill speed.
Step 1: Plot the 3 speeds on a number ...”

“Let B = # of boys in kindergarten
Let G = # of girls in kindergarten
Let J = price (in CENTS) of ONE juice box
Let S = price (in CENTS) of ONE soda
If every boy in a kindergarten class buys a soda and every girl in the same class buys a juice box, the class will spend 1¢ less in total than it ...”

“One option is to first solve the equation, 3x + 12 = 21, for x, and then determine the value of x+4.
Alternatively, we can factor the left side to get: 3(x + 4) = 21
Now divide both sides by 3 to get: x + 4 = 7
Answer: D
Cheers,
Brent”

“Some important rules:
#1. (ODD)(ODD) = ODD
#2. (ODD)(EVEN) = EVEN
#3. (EVEN)(EVEN) = EVEN
Given: a, b & c are integers
Target question: Is abc odd?
Statement 1: ab is odd
This tells us that a and b are both odd. However, we have no information about c.
Consider these two ...”

“One approach is to choose some values that meet the given information.
In the coordinate plane, the y-intercept of the line k is equal to twice its slope.
Okay, so let''s say the y-intercept of line k is 6. We get:
https://i.imgur.com/qJ5GeRQ.png
Since the y-intercept (6) is TWICE the ...”

“2 and 17 have no factors in common.
Since n must be divisible by both 2 and 17 -- and 2*17 = 34 -- n must be a MULTIPLE OF 34.
Thus, n must be divisible by I.
Eliminate any answer choice that does not include I.
Eliminate B and E.
If n=34, then n is not divisible by II or III.
Eliminate any ...”

“To keep track of the number of options for each position, draw a TREE.
Start with the MOST RESTRICTED position, which is SWEEPER.
If Sweeper = 3, then Goalie = 5.
If Sweeper = 4, then Fullback = 2.
Here''s the tree so far:
http://s11.postimage.org/laxnlsbof/soccer_half_tree.jpg
Now complete ...”

“A group...EACH has a team.
The wording above does not convey a logical meaning.
When each serves as an adverb, it cannot refer to a singular noun such as a group.
When each serves as an adverb, it may refer only to a PLURAL noun.
The prompt seems intended to convey the following:
Since ...”

“ Given: Madge paid A dollars for R bananas. Dean paid B dollars for S bananas.
Target question: Which one of them paid a higher price-per-banana?
Statement 1: B > A
This tells us that Dean PAID more than Madge paid.
However, we don''t know HOW MANY bananas each person bought.
...”

“Statement 1:
For every $100 commission that Dan earns, Jeff earns $120 in commission.
Since the commission rates for Dan and Jeff are unknown, no way to determine whether Jeff recorded more sales revenue than Dan.
INSUFFICIENT.
Statement 2:
Since Jeff''s sales and Dan''s sales can be any ...”

“Standard deviation describes how much the values in a set deviate from the mean. A larger standard deviation indicates that the values are deviating more -- getting farther away from -- the mean. So the question can be rephrased:
Do the values in set A deviate more from the mean than the do ...”

“x² < y² can be rephrased as follows:
|x| < |y|
Thus, the question stem can be rephrased as follows:
Is |x+10| < |y-10|?
|a-b| = the distance between a and b
|a+b| = |a-(-b)| = the distance between a and -b
Thus, the question stem can be further rephrased as follows:
Is the ...”

“First recognize that all radii (indicated with BLUE lines) have equal length
https://i.imgur.com/iz8ur0Z.png
Since BC and CD are both equal to the radius AB, we can see that all BLUE lines have equal length
https://i.imgur.com/mF0UxTp.png
This means the 2 blue triangles are EQUILATERAL ...”

“Target question: What distance did Jane travel?
Statement 1: Bill traveled 40 miles in 40 minutes.
No info about Jane
Statement 1 is NOT SUFFICIENT
Statement 2: Jane traveled at the same average rate as Bill
We don''t know Bill''s speed (rate)
Statement 2 is NOT SUFFICIENT
...”

“Target question: What is the measure of ∠E?
Given: Triangle DEF is an ISOSCELES triangle
This tells us that there are 2 equal angles
Statement 1: ∠D measures 42 degrees
Here it''s hard to tell whether there''s a second 42-degree angle or whether the other two angles are different. ...”

“If we aren''t wasting any wood, the length and width must be divisible by one side of the square
So, this question is a clever way of asking us what the greatest common divisor (GCD) of 18 and 30
The GCD of 18 and 30 is 6, so if we cut squares that are 6 x 6, then we won''t waste any wood.
We ...”

“The team has won 58 percent of the games it has played.
58% = 58/100 = 29/50.
The fraction in blue implies that the number of wins must be a MULTIPLE OF 29:
If 50 games are played, the number of wins = (29/50)(50) = 29.
If 100 games are played, the number of wins = (29/50)(100) = 58.
If 150 ...”

“The SC above is modeled -- not altogether successfully -- after SC30 in the OG12:
https://www.beatthegmat.com/not-sure-about-the-answer-of-a-question-from-og12-t265701.html
Rather than study the SC above, stick to the official SC.”

“The values in the chart DOUBLE-COUNT any student who belongs to 2 of the 3 clubs.
The totals for chess and drama -- 40 and 30 -- double-count the 10 students who belong to both chess AND drama.
The totals for chess and math -- 40 and 25 -- double-count the 5 students who belong to both chess AND ...”

“We want to determine how many integers end in 13
For example, 324713, 324813, 324913, 325013, . . . etc.
Recognize that, for every 100 consecutive integers, only 1 will end in 13.
There are 133,900 integers between 324,700 and 458,600 (since 458,600 - 324,700 = 133,900)
Of those 133,900 ...”

“This question could take a while to answer (especially if you don''t know where to begin! :-).
Fortunately, we can take advantage of an IMPORTANT feature about geometric diagrams on the GMAT:
The diagrams in GMAT problem solving questions are DRAWN TO SCALE unless stated otherwise.
We have ...”

“Take the task of building palindromes and break it into stages.
Stage 1: Select the ten-thousands digit
We can choose 1, 2, or 3
So, we can complete stage 1 in 3 ways
Stage 2: Select the thousands digit
We can choose 1, 2, or 3
So, we can complete stage 2 in 3 ways
Stage 3: Select ...”

“Let r=9, with the result that 1+r = 10.
Her husband can turn each dollar she gives him into 1+r, which will be deposited in a bank account.
Let the wife give the husband $1, implying that the husband turns this $1 into 1+r = $10.
Thus, the amount deposited into the account = $10.
The ...”

“Let T = the woman''s total salary for the year
During this year, she will give a fraction f of her salary to her husband, a private investor, to invest and they will live this year on the remainder.
So, the amount SAVED (and then given to huspand)= fT
This means the amount SPENT = T - fT ...”

“IF all 4 sides of the rectangle have EQUAL length (i.e,, the shape is a square), then the diagonal will create two 45-45-90 right triangles.
However, if the sides are not all the same length, the resulting triangles will NOT be 45-45-90 right triangles.
Cheers,
Brent”

“ Given: A father distributed his total wealth to his two sons.
Target question: How much wealth did the father have?
Statement 1: The elder son received 3/5 of the wealth.
Without any dollar values, there''s no way to answer the target question with certainty.
So, statement 1 is NOT ...”

“IMPORTANT: The two statements essentially provide the exact same information.
If it took Erik 2 hours to travel 10 miles, we can conclude that his average speed was 5 miles per hour.
When the two statements provide the same information in a Data Sufficiency question, the correct answer must be ...”

“Here are a few similar questions to practice with:
- https://www.beatthegmat.com/if-is-the-greatest-integer-less-than-or-equal-to-x-what-t302459.html
- https://www.beatthegmat.com/g-x-is-the-greatest-integer-less-than-or-equal-to-x-t298237.html
- ...”

“Let the distance between A and B = 120 miles.
Since the 120-mile distance between A and B is twice the distance between B and C, the distance between B and C = 60 miles, implying that the total distance = 120+60 = 180 miles.
Since the average speed for the 120-mile leg between A and B = 60 ...”

“We can also solve the question using 1 variable
Norms photos number 10 more than twice the number of Lisa''s photos
Let x = the number of photographs that Lisa took.
So, 2x + 10 = the number of photographs that Norm took.
The total of Lisa and Mikes photos is 50 less than the sum of ...”

“Let''s assign a "nice" value to the number of widgets made by ONE employee.
We''ll also assign a nice value to the number of employees on each shift.
At a certain factory, each employee working the SECOND shift produced 2/3 as many widgets as each employee working the FIRST shift.
...”

“The sum for Lisa and Mike is 50 less than the sum for Norm and Mike.
Since Mike is common to both sums, the 50-photo difference must be yielded solely by Lisa and Norm.
Implication:
Lisa took 50 fewer photos than Norm.
We can PLUG IN THE ANSWERS, which represent the number of photos taken ...”

“Let h=3.
Since an average of $78 is sold per hour, the total revenue for 3 hours = 3*78 = 234.
Since the cost for the day is composed of $150 for the table rental and $10 per hour for the assistant, the cost for the day = 150 + 3*10 = 180.
P = profit = revenue - cost = 234 - 180 = 54.
The ...”

“It takes for granted that X implies that X is an ASSUMPTION: a statement that must be true for the conclusion to hold.
Thus, when X is negated, the conclusion must be invalidated.
B: It takes for granted that the volunteers with the highest levels of self-esteem had better social skills than did ...”

“Premise:
Volunteers with the highest levels of self-esteem consistently rated themselves as having much better social skills than did the volunteers with moderate levels.
Conclusion:
Attaining an exceptionally high level of self-esteem greatly improves one’s social skills.
The correct answer ...”

“We''ll begin by arbitrarily placing point A somewhere on the circumference.
https://i.imgur.com/hRHtq4o.png
So, we want to know the probability that a randomly-placed point B will yield a chord AB that is at least 2 cm long.
So, let''s first find a location for point B that creates a chord ...”

“These kinds of questions (Variables in the Answer Choices - VIACs) can be answered algebraically or using the input-output approach.
Mitch has solved the question using the input-output approach, so let''s use the algebraic approach.
2k years ago Frank was 3k years old.
We''re given Frank''s ...”

“Let k=10.
2k years ago Frank was 3k years old.
Since k=10, 20 years ago Frank was 30 years old, implying that his age now = 30+20 = 50.
In k years Frank''s age, in years, will be:
Since Frank is now 50 years old, Frank''s age in 10 years = 50+10 = 60.
The correct answer must yield a value ...”

“Alternate approach:
For any quadratic in the form x² + bx + c = 0, where b and c are rational numbers:
If one of the roots is m+√n, then the other root is m-√n.
Sum of the roots = -b
Product of the roots = c
Here, since one of the roots is 1+√2, the other root is 1-√2.
Thus:
-b ...”

“In order to minimize the number of homeowners, we must MAXIMIZE the number of men in the group, since the proportion of male homeowners (20%) is less than the proportion of female homeowners (25%)
So, let''s see what happens if there are 199 men and 1 woman.
If 20% (aka 1/5) of the men are ...”

“NOTE: Always SCAN the answer choices before performing any calculations.
Here, we notice that the answer choices are pretty small, so if we don''t come up a full solution, we should consider LISTING AND COUNTING
We get the following extensions:
1236
1326
1362
1632
2136
2316
3126
3162
...”

“Take the task of arranging the 4 digits and break it into stages.
We’ll begin with the most restrictive stage.
Stage 1: Select the digit in the units position
Since the 4-digit extension must be EVEN, the unit digit must be either 2 or 6
So, we can complete stage 1 in 2 ways
Stage ...”

“Target question: What was the revenue that a theater received from the sale of 400 tickets
Jump straight to . . .
Statements 1 and 2 combined
The combined statements tell us that 100 tickets sold for the full price of $25 each.
The remaining 300 tickets were sold at some reduced price.
...”

“Another approach is to use the Double Matrix Method. This technique can be used for most questions featuring a population in which each member has two characteristics associated with it (aka overlapping sets questions).
Here, we have a population of students, and the two characteristics are:
- ...”

“In any given year, there will be 52 or 53 Mondays, 52 or 53 Tuesdays, etc.
So, it''s impossible to say that it rained on 40% of all Mondays or 20% of all Tuesdays, etc.
For example, if there are 52 Mondays, then 40% of 52 = 20.8, which means that it rained on 20.8 days (impossible).
Cheers, ...”

“Since each payday is $10 more than the preceding payday, the list of 15 paydays constitutes an EVENLY SPACED SET.
Statement 1:
For any evenly spaced set:
Median = Sum/Quantity.
Thus:
Pay on the 14th day = median of the 3 most recent days = sum/quantity = 690/3 = 330.
Since the pay on the ...”

“Original deposit: 10,000.
The balance doubles every 10 years.
After 10 years, we get:
2*10,000 = 20,000.
After 20 years, we get:
2*20,000 = 40,000.
After 30 years, we get:
2*40,000 = 80,000.
The correct answer is E.”

“Let''s see what happens if a man waits until the last (5th) round to take the money.
Round #1: $1 placed in front of the man (but he doesn''t take it)
Round #2: Now, $2 placed in front of the man (but he doesn''t take it)
Round #3: Now, $4 placed in front of the man (but he doesn''t take it)
...”

“Target question: Is a% of b% of c an integer?
This is a great candidate for rephrasing the target question.
Aside: We have a free video with tips on rephrasing the target question:
a% of b% of c is the same as (a/100)(b/100)(c), which equals abc/10,000
So, we can rephrase the target ...”

“The number of three-member committees that can be formed from a group of nine members
Since the order in which we select the 3 people does not matter, we can use combinations.
We can choose 3 people from 9 people in 9C3 ways
9C3 = (9)(8)(7)/(3)(2)(1) = 84 three-member committees
The total ...”

“Let the 6 vertices = A, B, C, D, E and F.
P(the triangle includes AT LEAST 1 side of the hexagon) = 1 - P(the triangle includes NO sides of the hexagon).
From the 6 vertices, the number of ways to choose 3 to form a triangle = 6C3 = (6*5*4)/(3*2*1) = 20.
Of these 20 triangles, only 2 include ...”

“Here''s another approach.
Take the task of selecting the 3 committee members and break it into stages.
Stage 1: Select the 3 twins from which we will select 1 spouse each.
There are 4 sets of twins, and we must select 3 of them. Since the order in which we select the 3 twins does not ...”

“Number of options for the 1st person = 8.
Number of options for the 2nd person = 6. (Of the 7 people left, we can''t use the twin of the 1st person chosen, leaving 7-1= 6 choices.)
Number of options for the 3rd person = 4. (Of the 6 people left, we can''t use the twins of the 2 people already ...”

“Saleswoman receives a commission equal to 20 percent of the first $500 of the total amount of the sale, plus 30 percent of the total amount in excess of $500
If she sold $800, then...
Commission = (20 percent of $500) + (30 percent of $300)
= $100 + $90
= $190
The saleswoman''s commission ...”

“Height of tree on day 0 = 4
Let d = the height increase each year
Height of tree at the end of the 1st year = 4+d
Height of tree at the end of the 2nd year = 4+d+d = 4 + 2d
Height of tree at the end of the 3rd year = 4+d+d+d = 4 + 3d
Height of tree at the end of the 4th year = 4+d+d+d+d = 4 + ...”

“Let x = the yearly increase.
Height after 4 years = 4 + 4x.
Height after 6 years = 4 + 6x.
At the end of the 6th year, the tree was 1/5 taller than it was at the end of the 4th year.
The phrase in red implies the following:
The height in the 6th year was 6/5 of the height in the 4th ...”

“ Given: Last Friday a certain shop sold 3/4 of the sweaters in its inventory. Each sweater sold for $20.
Target question: What was the total revenue last Friday from the sale of this sweaters?
Statement 1: When the shop opened last Friday, there were 160 sweaters in its inventory.
3/4 ...”

“The long approach is to meticulously convert 11/25 of a minute to seconds, and then ADD that unaccounted time to Nuria''s recorded time (of 5 minutes, 44 seconds)
However, when we check the answer choices before performing any calculations (ALWAYS check the answer choices before performing any ...”

“Any integer that is a multiple 10 is also a multiple of 5.
Implication:
Any integer that satisfies Statement 1 will also satisfy Statement 2.
Both statements are satisfied by the following cases:
Case 1: p=10, which is a multiple of 10 (and thus is also a multiple of 5)
In this case, p is NOT a ...”

“The drawings below satisfy both statements:
http://s4.postimage.org/1yjxdhskk/ABCD_number_line.jpg
Since in the first case BD=1 and in the second case BD=17, the two statements combined are insufficient.
The correct answer is E.”

“Case 1: Team A wins a prize, with the result that Team B must win a prize
Number of prizes that could be won by A = 3.
Number of remaining prizes that could be won by B = 2.
Number of teams that could win the third prize = 4. (Any of the 4 remaining teams.)
To combine these options, we ...”

“Let n=50, implying that 50% of the sold juice packs are A, while the other half are B.
In other words, the store sells an EQUAL NUMBER of each type of juice pack.
Let the store sell 2 OF EACH TYPE of juice pack.
Revenue from two $1 packs of A = 2*1 = 2.
Revenue from two $1.5 packs of B = ...”

“A student asked me to answer this question, so here it goes . . .
1) Read the question stem to determine the question type.
Tricky!
With fill-in-the-blank questions, it''s often difficult to determine the question type.
The last sentence reads Unitron is not going to hire department heads ...”

“Let James'' rate = 3 mph.
James started from his home and drove eastwards at a constant speed. Exactly 90 minutes after James stated from his home, his brother Patrick started from the same point and drove in the same direction as James did at a different constant speed. Patrick overtook James ...”

“The equation for a circle with its center at the origin and a radius of r is as follows:
x² + y² = r².
Thus, x² + y² = 25 is a circle with its center at the origin and a radius of 5.
A point on this circle will have integer coordinates in three cases:
Case 1: The radius is horizontal
...”

“Target question: Is the two-digit integer xy prime?
Given: Each of the two digits x and y is distinct
Statement 1: Each of the digits x and y is the sum of 2 distinct single digit prime numbers.
Let''s TEST some values.
There are several values of x and y that satisfy statement 1. Here ...”

“Target question: How many apples did Joan buy?
Given: Joan spent $10 to buy at least one piece each of apples and oranges at a store where each apple cost $2 and each orange cost $1.
Since there are only a handful of possible outcomes, we might benefit from quickly listing them:
i) Joan ...”

“This question is somewhat ambiguous, since it''s hard to say whether "nondrinkers" in statement 1 refers to people who drank neither beer nor wine, or did not drink beer. I''m assuming that it means to not drink beer, so I''ve add that to the question below.
One approach is to use ...”

“Target question: What is the hundreds digit of the product AB?
Given: The difference between positive two-digit integer A and the smaller two-digit integer B is twice A‘s units digit
Let x = the tens digit of A, and let y = the units digit of A
So, the VALUE of A = 10x + y
From the ...”

“Target question: What is the probability that the child will be a boy?
This is a good candidate for rephrasing the target question.
Let G = # of girls in the school
Let B = # of boys in the school
So, G + B = total number of children in the school
So, P(selected child is a boy) = B/(G + B)
...”

“Take the task of selecting the president and two vice-presidents and break it into stages.
Stage 1: Select the president
There are 10 people to choose from. So, we can complete stage 1 in 10 ways
Stage 2: Select two people to be the vice-presidents
Since the order in which we select the ...”

“Let''s PLUG IN a nice value for the total distance traveled.
If Derek''s average speed is 2.8 mph, then let''s say that he traveled a total of 28 miles.
At an average rate of 2.8 mph, a 28 mile trip will take 10 hours.
Since Derek''s average speed is BETWEEN 2 and 3 mph, we can conclude that ...”

“The average speed -- 2.8 miles per hour -- must be BETWEEN the two individual rates (s and s+1).
Thus, s = 2 miles per hour and s+1 = 3 miles per hour.
We can PLUG IN THE ANSWERS, which represent the fraction traveled at 2 miles per hour.
When the correct fraction is plugged in, the average ...”

“Number of options for president = 10. (Any of the 10 members.)
From the remaining 9 members, the number of ways to choose 2 to serve as vice-president = 9C2 = (9*8)/(2*1) = 36.
To combine these options, we multiply:
10*36 = 360.
The correct answer is E.”

“When it comes to remainders, we have a nice rule that says:
If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), ...”

“Target question: Is b even?
KEY POINT: We are not told that a and b are INTEGERS.
As such, we can jump straight to. . . .
Statements 1 and 2 combined
There are several values of a and b that satisfy BOTH statements. Here are two:
Case a: a = 4 and b = 2. Notice that a/b = 4/2 = 2, ...”

“Target question: Does line k intersect quadrant II?
Statement 1: The slope of k is -1/6
Here are a few lines with slope -1/6
https://i.imgur.com/1p6USLh.png
KEY CONCEPT: As we travel from right to left along a line with slope -1/6, we keep moving up.
So, at some point, the line will ...”

“Judging from the OA, I''ve added some key text below
A certain recipe calls for 2 cups of sugar and 3 1/2 cups of flour
So sugar/flour = 2/3.5
Check answer choices . . . . not there.
Looks like we need to create a ratio that''s EQUIVALENT to 2/3.5
Multiply top and bottom by 2 to get ...”

“APPROACH#2
When the positive integer x is divided by 6, the remainder is 4
In other words, x is 4 greater than some multiple of 6
We can write: x = 6k + 4, where k is an integer
If x = 6k + 4, we can also write: x = 6k + 3 + 1
Factor to get: x = 3(2k + 1) + 1
Since 2k+1 must be an integer, ...”

“----ASIDE-------------------
When it comes to remainders, we have a nice rule that says:
If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, ...”

“A chili recipe calls for the final mixture to consist of 1/2 meat and 1/3 beans, with the remainder an even mixture of sauce and various spices
Let''s see what fraction of the final mixture is sauce and various spices
1/2 + 1/3 = 3/6 + 2/6 = 5/6
If 5/6 of the mixture is meat and beans, then the ...”

“ Given: All the branded shirts are priced at $60 per unit and all the non-branded shirts are priced at $20 per unit. On a certain day, the store sold a total of 30 shirts.
Let B = # of branded shirts sold
Let N = # of non-branded shirts sold
If a TOTAL of 30 shirts were sold, we can write: N + ...”

“Target question: What was the total number of female voters who voted for Resolution K?
When solving Data Sufficiency questions, it''s important that you don''t perform more calculations than are necessary. So, for this question, we need only determine whether or not we have sufficient ...”

“The above solutions are great.
I just wanted to add that whenever you''re told that a value is non-negative on the GMAT, you should be sure to consider what happens when that value equals zero
Keep in mind that, if the test-makers wanted to restrict a value to POSITIVE values only, then they''d ...”

“IMPORTANT CONCEPT: The prime factorization of a perfect square (the square of an integer) will have an even number of each prime
For example: 400 is a perfect square.
400 = 2x2x2x2x5x5. Here, we have four 2''s and two 5''s
This should make sense, because the even numbers allow us to split the ...”

“The key here is MUST be true.
So, if we can find an example in which a statement is NOT true, then we can ELIMINATE it.
We have:
Set P {a, b, c, d, e, f, g}
Set Q {a, b, c, d, e, f}
I''ll make the extra value red so it''s easier to compare the two sets
So, the sets COULD be:
Set P {1, 2, ...”

“Target question: How many objects are in the box?
Given: Every object in a box is either a sphere or a cube, and every object in the box is either red or green.
We can solve this using the Double Matrix Method.
This technique can be used for most questions featuring a population in which ...”

“One approach is to use the following equation:
(machines)(time) / output = (machines)(time) / output
In the equation above:
Machines and time are INVERSELY PROPORTIONAL.
As the number of machines increases, the amount of time required to produce the same output decreases.
Machines and ...”

“One approach:
Good Cases = Total Cases - Bad Cases.
Total Cases:
From 9 points, the number of ways to choose 3 = 9C3 = (9*8*7)/(3*2*1) = 84.
Bad Case 1: Choosing 3 collinear points from the 5-point line, with the result that a triangle cannot be formed
From the 5 points on this line, the ...”

“Choose a "nice" number for the sales revenue
Let $100 = LAST MONTH''s revenue
So, $120 = THIS MONTH''s revenue (since sales were up 20 percent from last month)
LAST month, a florist gave 7 percent of her sales revenue to her employees in bonuses
So, bonuses = 7% of $100
= $7
...”

“There are two ways in which we can create a triangle.
#1) Select 2 points from the 5-point line and select 1 point from the 4-point line.
#2) Select 2 points from the 4-point line and select 1 point from the 5-point line.
#1) Select 2 points from the 5-point line and select 1 point from the ...”

“ Given: Alan is NOW 3 miles ahead of Peter
Target question: How many minutes from now will Peter be 1 mile ahead of Alan?
Statement 1: ONE HOUR AGO, Alan was 7 miles ahead of Peter.
Alan is NOW 3 miles ahead of Peter
So, in one hour, the GAP between Alan and Peter decreased by 4 miles
...”

“Let''s assign a "nice" value to the job, a value that works well with the given values (6 days and 4 days ).
So, let''s say the ENTIRE job is to make 24 widgets
R can complete a certain job in 6 days
In other words, R can make 24 widgets in 6 days
So, R can make 4 widgets per day
...”

“Let x = the duration of the phone call (in minutes)
Under plan A, the company charges a total of $0.60 for the first 7 minutes of each call and $0.06 per minute thereafter.
So, the TOTAL cost = $0.60 + (x - 7 minutes)($0.06)
= 0.60 + (0.06x - 0.42)
= 0.06x + 0.18
Under plan B, the company ...”

“Since the prompt indicates that n is an integer greater than 1, the case in red is not possible.
The only value that satisfies Statement 2 is n=2, making Statement 2 SUFFICIENT.
The correct answer thus is not C but B.”

“Between 1982 and 1987 are 4 non-leap years and 1 leap year.
Each non-leap year is composed of 365 days; a leap year is composed of 366 days.
Thus, from June 25, 1982 to June 25, 1987, the total number of days = 4(365) + 366.
Since June 25, 1982 is a Friday, every passing of 7 days will yield ...”

“(man''s rate)/(woman''s rate) = (4 mph)/(20 mph) = 1/5.
Between the moment when the woman passes the man and the moment when she stops to wait, the woman and the man each travel for 5 minutes.
During these 5 minutes, the woman travels x miles.
Since the man''s rate is 1/5 the woman''s rate, the ...”

“Target question: Does the line intersect quadrant III?
Jump straight to...
Statements 1 and 2 combined
Here are two scenarios that satisfy BOTH statements:
Case a: m = 3
https://i.imgur.com/2SVui9E.png
In this case, the answer to the target question is YES, the line DOES intersect ...”

“Target question: What was the value of y?
Statement 1: The revenue for the website that year was $36,000.
First round of "investors": y people paying $30 each = 30y dollars
Second round of "investors": y² people paying $15 each = 15y² dollars
Total revenue = 15y² + ...”

“Statement 1:
Here, n can be any prime number.
INSUFFICIENT.
Statement 2:
(n+2)/n = n/n + 2/n = 1 + 2/n.
Given that n must be an integer, 1 + 2/n will yield an integer value only if n is a FACTOR OF 2.
Factors of 2: 1 and 2.
Since n must be greater than 1, n=2.
SUFFICIENT.
The correct ...”

“Target question: What is the ratio of alcohol to pigment in Luxo paint?
Given: Luxo paint contains ONLY alcohol and pigment.
Statement 1: Exactly 7 ounces of pigment are contained in a 12-ounce can of Luxo paint.
If 7 of the 12 ounces of paint are PIGMENT, then the remaining 5 ounces ...”

“ Given: Tickets for children cost $10 and tickets for adults cost $35. Ticket revenues from the last performance were $390
Let A = # of adults in attendance
Let C = # of children in attendance
So, we can write: 35A + 10C = 390
Since A and C must be POSITIVE INTEGERS, there are not many ...”

“When it comes to remainders, we have a nice rule that says:
If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), ...”

“We can also solve the question by testing the answer choices
Here''s what I mean:
The diagram tells us that x= 0 is a solution to the inequality.
When we check the answer choices, we see that answer choice D (1 < 2x < 3) does NOT have x = 0 as a solution.
When we let x = 0, the ...”

“Target question: What is the value of n?
Given: n is an integer between 30 and 50 inclusive
Statement 1: When n is divided by 8, the remainder is 7
------ASIDE--------------------------------------
When it comes to remainders, we have a nice rule that says:
If N divided by D leaves ...”

“Let''s work BACKWARDS.
Let''s say the WHOLESALE price was $100
the merchant made a net profit of 20% on the wholesale cost (which was $100)
So, the merchant sold the sweater for $120
Let M = the markup price on the label (before any discounts).
So, 40% less than M = $120
In other ...”

“Statement 1:
Rate and time have a RECIPROCAL relationship.
A rate ratio of (40kph) : (50kph) = 4:5 implies a time ratio of 5:4 = 10:8.
The time ratio in blue implies that -- for the time at 40kph to be 2 hours more than the time at 50kph -- the time at 40kph must be 10 hours, while the time at ...”

“Target question: How many bottles of beer did Kevin buy?
Given: Kevin pays $1.00 for each can of beer and $1.50 for each bottle of beer. Kevin buys a total of 15 bottles and cans of beer
Let C = the NUMBER of Cans that Kevin bought
Let B = the NUMBER of Bottles that Kevin bought
So, we can ...”

“Statement 1:
The TIME RATIO for Tim and Glenn = 48:60 = 4:5.
Since time and rate have a RECIPROCAL relationship, the RATE RATIO for Tim and Glenn = 5:4.
Implication:
For every 5 laps that Tim runs, Glenn runs only 4 laps.
Thus, after running 5 laps, Tim will have run 1 more lap than Glenn. ...”

“A retail appliance store priced a video recorder at 20 percent above the wholesale cost of $200
20% of $200 = $40
So, the retail price = $200 + $40 = $240
If a store employee applied the 10 percent employee discount to the retail price to buy the recorder, how much did the employee pay for the ...”

“It''s a good idea to first try to understand what the graph is telling us.
For example, the leftmost point on the curve has the coordinates (10, 340)
This tells us that, if the length of the playground is 10 meters, then a total of 340 meters of fencing is required.
This makes sense, since we ...”

“Height of tree on day 0 = 4
Let d = the height increase each year
Height of tree at the end of the 1st year = 4+d
Height of tree at the end of the 2nd year = 4+d+d = 4 + 2d
Height of tree at the end of the 3rd year = 4+d+d+d = 4 + 3d
Height of tree at the end of the 4th year = 4+d+d+d+d = 4 + ...”

“In a rhombus, the diagonals are perpendicular bisectors.
http://i1168.photobucket.com/albums/r500/GMATPrepNow/rhmbs1_zps1nyufj6q.png
So, we can add our lengths as follows.
http://i1168.photobucket.com/albums/r500/GMATPrepNow/rhmbs2_zpskyv3wwkl.png
From here, if we focus on one of the 4 ...”

“Let the one-bedroom rent = $2 per month, implying that the two-bedroom rent = 2 + (50% of 2) = 2 + 1 = $3 per month.
Since the rent for a one-bedroom apartment = $2 per month, the total amount to rent a one-bedroom apartment for 12 months = 2*12 = $24.
For $24, the number of months yielded for a ...”

“NOTE: We can write the ratio x:y as a fraction x/y
Target question: What is the value of x/y ?
Statement 1: 3x = 4y
Divide both sides by y to get: 3x/y = 4
Divide both sides by 3 to get: x/y = 4/3
So, the answer to the target question is x/y = 4/3
Since we can answer the target question ...”

“A fast and easy approach is to assign a "nice" value to the monthly rent of a ONE-bedroom apartment.
Let''s say the rent on a ONE-bedroom apartment is $100 per month
A two-bedroom apartment costs 50% more to rent than a one-bedroom apartment
50% of $100 = $50
So, the rent on a ...”

“ Given: Line k has the equation Ax + By = C
Target question: What is the slope of line k ?
This is a good candidate for rephrasing the target question.
Let''s take the given equation Ax + By = C and rewrite it in slope y-intercept form, y = mx + b, where m = slope and b = y-intercept
...”

“Target question: What is the ratio of x:y:z?
Statement 1: xy = 14
In information about z
Statement 1 is NOT SUFFICIENT
Statement 2: yz = 21
In information about x
Statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
There are several values of x, y and z that satisfy BOTH ...”

“Let W = Winesap crates, M = McIntosh crates, and R = Roma crates.
Since a total of 25 crates are ordered, we get:
W + M + R = 25.
To MINIMIZE the value of W, we must MAXIMIZE the values of M and R.
Since the store is to order more crates of Winesap than crates of McIntosh and more crates of ...”

“The prompt states:
Every day, the two machines operate constantly from 08:00 to 18:00.
As shown in my post above, Statement 2 yields the following rate ratio:
(W alone) : (W and D together) = 1:5.
The rate ratio indicates that -- of every 5 kg produced when W and D work together from 8:00 to ...”

“Generally, for + VERBing is an ADJECTIVE serving to modify a NOUN.
A: The Breeding Bird Survey uses annual roadside counts along established routes for monitoring.
Here, for monitoring seems to be an ADJECTIVE modifying routes, implying that the ROUTES were designed FOR MONITORING.
Not the ...”

“Since we have an ODD number of values, the median will be the MIDDLEMOST value. In other words, if we arrange all 73 values in ASCENDING order, the median will be the 37th value (i.e., 36 values are below and 36 values are above).
The table says the first 2 scores in the 50-59 range. So, the 1st ...”

“Let''s find some points that lie on each of the curves.
So, for each equation, we''ll find a pair of values (an x-value and a y-value) that satisfy each equation.
We''ll do so by plugging in some x-values and calculating the corresponding y-values.
Let''s start with x = 0
Plug x = 0 into ...”

“Let x = the two-digit number.
Since dividing by 4 leaves a remainder of 1, x is equal to one more than a multiple of 4:
x = 4k + 1
Since dividing by 14 leaves a remainder of 1, x is equal to one more than a multiple of 14:
x = 14m + 1
Since the two expressions in blue must be equal, we ...”

“When it comes to remainders, we have a nice rule that says:
If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), ...”

“Here''s a step-by-step algebraic solution.
Let T = the TOTAL value of the item.
Leo paid a 7% import tax on the portion of the total value of the item in EXCESS of $1000
So, Leo pays tax on the amount that''s GREATER then $1000
So, Leo pays 7% tax on (T - 1000)
We can write: import tax = ...”

“Statement 1: If the selling price per coat had been twice as much , the store’s gross profit on the 20 coats would have been $2,400
When the selling price per coat is doubled, the resulting profit per coat = (total profit)/(number of coats) = 2400/20 = 120.
Case 1: Double the selling price = ...”

“60! = 60*59*58*....*3*2*1.
Since 10=2*5, EVERY COMBINATION OF 2*5 contained within the prime-factorization of 60! will yield a 0 at the end of the integer representation of 60!.
The prime-factorization of 60! is composed of FAR MORE 2''S than 5''s.
Thus, the number of 0''s depends on the ...”

“An INSCRIBED ANGLE is formed by two chords.
A CENTRAL ANGLE is formed by two radii.
When an inscribed angle and a central angle intercept the same two points on a circle, the inscribed angle is 1/2 the central angle.
https://i.postimg.cc/qgbxWKdW/Angle-CEB-1.png
Here, inscribed angle CFB and ...”

“P(selecting a sibling pair) = P(select a junior with a sibling AND select the senior who is that junior''s sibling)
= P(select a junior with a sibling) x P(select the senior who is that junior''s siblingx 1/800
= 60/800,000
= 3/40,000
= A
Note: P(select a junior with a sibling) = 60/1000, ...”

“Let''s use some useful circle properties
First let''s add a blue line, to divide ∠CEB into 2 angles
https://i.imgur.com/xe05QIU.png
CD is the DIAMETER of the circle
Since ∠CED is an inscribed angle containing (aka "holding") the diameter, we can conclude that ∠CED = 90°
...”

“Target question: Is the tallest child in Class A is taller than tallest child in Class B?
Jump to . . .
Statements 1 and 2 COMBINED
There are several scenarios that satisfy BOTH statements. Here are two:
Case a: Class A heights = {1, 50} and Class B heights = {2, 40}. In this case, the ...”

“When it comes to probability questions involving "at least," it''s best to try using the complement.
That is, P(Event A happening) = 1 - P(Event A not happening)
So, here we get: P(getting at least 1 man) = 1 - P(not getting at least 1 man)
What does it mean to not get at least 1 man? ...”

“Here''s a somewhat long approach (but it works!):
Target question: What is the remainder when n is divided by 6?
Given: Positive integer n is greater than 6
Statement 1: When n is divided by 9, the remainder is 2.
------ASIDE----------------
When it comes to remainders, we have a ...”

“Statement 1: x=12u, where u is an integer and x=8y+12.
In other words, x is a multiple of 12.
For x to be a multiple of 12, 8y must be a multiple of 12.
If y=3, then x = 8*3 + 12 = 36.
The GCD of 3 and 36 is 3.
If y=6, then x = 8*6 + 12 = 60.
The GCD of 6 and 60 is 6.
Since the ...”

“Target question: What is the greatest common divisor of x and y?
Given: x = 8y + 12
Statement 1: x = 12u, where u is an integer.
There are several pairs of values that satisfy the given conditions. Here are two:
Case a: x=36 and y=3, in which case the GCD of x and y is 3
Case b: x=60 ...”

“As with many probability questions, we can also solve this using counting techniques.
P(2 middle are red) = (# of outcomes with 2 red in middle)/(total number of outcomes)
Label the 4 bushes as W1, W2, R1, R2
total number of outcomes
We have 4 plants, so we can arrange them in 4! ways = ...”

“ Given: Of the first three dozen (36) cookies baked at a bake shop on a certain day, 1/3 were chocolate chip cookies. 1/2 of the remaining cookies that were baked that day were chocolate chip cookies,
1/3 of 36 = 12. So, 12 chocolate cookies were baked in the first round of baking.
Let x = ...”

“From the time the first paperback copy was sold until the last copy of the novel was sold 9 times as many paperback copies as hardback copies were sold
Let x = number of hardback copies sold during this time
So 9x = number of paperback copies sold during this time
So, x + 9x = total number of ...”

“Time and rate have a RECIPROCAL RELATIONSHIP:
2 times as fast implies 1/2 the time.
3 times as fast implies 1/3 the time.
Let t = the normal time.
Since the bus travels today at 7/6 its normal rate, the time today is equal to 6/7 of the normal time:
(6/7)t.
Since the bus leaves 1/2 hour late ...”

“Let x = number of QUARTERS in pocket
So, 0.25x = VALUE of quarters in pocket
He has three times as many dimes as quarters
So, 3x = number of DIMES in pocket
So, 0.10(3x) = 0.3x = VALUE of dimes in pocket
He has six times as many nickels as dimes.
So, (6)(3x) = number of NICKELS in ...”

“I created this question to highlight many students'' tendency to avoid listing and counting as a possible approach.
As you''ll see, the approach is probably the fastest approach.
P(no one receives his own hat) = (# of outcomes in which no one receives his own hat)/(TOTAL number of outcomes)
...”

“Hello Beat the GMAT users,
Finding the right business programs can be a challenge. There are so many choices. :? Thats why we want to help you navigate this journey and unlock your best-fit opportunities with personalized business schooll recommendations. :) These recommendations are based on ...”

“Yes, great work, Mitch!
I thought I''d mention that, once we know the correct answer is a multiple of 222 (which is also a multiple of 3), then the correct answer will be a multiple of 3.
Useful property: If a number is a multiple of 3, then the sum of its digits is also a multiple of 3.
When we ...”

“I created this question to show that there can be times when the best (i.e., fastest) way to solve a counting question is by listing and counting
How do we know when it''s not a bad idea to use listing and counting?
The answer choices will tell us (ALWAYS scan the answer choices before ...”

“Target question: How many men are in a certain company''s vanpool program?
Statement 1: The ratio of men to women in the program is 3 to 2.
Let''s TEST some values.
There are several scenarios that satisfy statement 1. Here are two:
Case a: There are 3 men and 2 women in the program. In ...”

“Given any set that is symmetrical about the median:
Sum = (quantity)(median)
The set of 3-digit integers that can be formed from the digits 1,2 and 3 is symmetrical about the median (222):
...212, 213, 221, 222, 223, 231, 232...
Thus:
Sum of the 27 integers = (quantity)(median) = 27*222 = ...”

“GIVEN: When the men start walking, Brian has a 30-mile lead
Let B = Brian''s walking speed (in miles per hour)
Let A = Ashok''s walking speed (in miles per hour)
Since Ashok''s speed is greater than Brian''s speed, the rate at which the gap shrinks = (A - B) miles per hour
For example, if A ...”

“If you go here (https://www.gmatprepnow.com/content/improvement-chart) and download the Improvement Chart for the OG2016, you''ll see the question breakdown for each topic.
This should help.
Cheers,
Brent”

“This question begs for some prime factorization.
88,000 = (2)(2)(2)(2)(2)(2)(5)(5)(5)(11)
First, we can see that there must be one (11-point) red chip.
Now, what role do these 2''s play? Since there are no 2''s hiding among the 5-point chips or the 11-point chips, the 2''s must be associated ...”

“GIVEN:
For example, [3.2] = 3, since 3 is the greatest integer that is less than (or equal) to 3.2
Likewise, [8.7] = 8, since 8 is the greatest integer that is less than (or equal to) 8.7
Now onto the question....
√25 = 5
√36 = 6
√49= 7
√64 = 8
Since √30 is BETWEEN √25 ...”

“For the purposes of a Y/N DS problem, the rephrase is the equivalent of the original question stem.
Original question stem: Is x² > xy?
The answer will be YES if y/x is less than 1..
The answer will be NO if y/x is NOT less than 1.
Thus, the original question stem can be rephrased as ...”

“Target question: Is |1 - 4k| < k?
Statement 1: k > 4x³
This pretty much tells is that k can have ANY value.
For example, notice that, if x = -100, then 4x³ = -4,000,000
So, for this value of x, k can be any number greater than -4,000,000
Let''s TEST some values.
There are ...”

“Target question: Which sport utility vehicle has a higher list price, the Touristo of the Leisure?
Statement 1: The list price of the Leisure is 5/6 the list price of the Touristo.
If the Leisure''s price is 5/6 of the Touristo''s price, then the answer to the target question is the Touristo ...”

“x² > xy is true only if x is NONZERO, in which case we can safely divide by x², since the square of a nonzero value must be positive:
x²/x² > (xy)/x²
1 > y/x
Question stem, rephrased:
Is y/x < 1?
Statement 1: x > y
If x=2 and y=1, then y/x =1/2. so the answer to the ...”

“Let the job = the LCM of the 3 times = 60 widgets.
Maximum time:
The maximum time is yielded when the slowest person -- R -- works alone.
Since R takes 15 hours to complete the 60-widget job, the maximum time = 15 hours.
Minimum time:
The minimum time is yielded when all 3 people work ...”

“i) When x is divided by y, the remainder is 2x
The remainder cannot be greater than the dividend (the number we''re dividing)
For example, it CANNOT be the case that 17 divided by 5 leaves a remainder of 34
Statement i can never be true
Check the answer choices. . . . ELIMINATE A and D
ii) ...”

“X led to Y implies that X did not directly cause Y but that X was a factor that helped to yield Y.
SC755 in the OG17:
An inability to adapt led to their extinction.
Here, an inability to adapt did not directly cause the extinction but was a factor that helped to yield the extinction.
A cause ...”

“Car K travels at a constant speed that is 80% the constant speed of Car J
Let''s assign some nice values to the speeds.
Let''s say Car J travels 40 miles per hour
So, Car K travels 32 miles per hour
Car J departs from City A 15 minutes after Car K does
Let''s calculate the distance that Car ...”

“Generally, COMMA + as-clause implies that the clause after as LEADS to the preceding clause, with the result that the two clauses happen concurrently.
An OA in GMATPrep:
The steel industry has changed radically over the last two decades, as large, integrated companies such as Bethlehem Steel have ...”

“Let B = Barry''s rate and W = the walkway''s rate.
The distance can be ANY VALUE.
Let the distance = 240 meters.
WITH the walkway, the time = 30 seconds:
Here, Barry and the walkway WORK TOGETHER, so we ADD their rates:
B+W = d/t = 240/30 = 8 meters per second.
AGAINST the walkway, the ...”

“Let''s use the INPUT-OUTPUT approach.
It might be useful to choose a number that works well with the fractions given in the question (1/3 and 1/7).
So, let''s say there are 21 family members at the party.
In other words, we''re saying that x = 21
1/3 chose to have a hamburger, and of ...”

“Let the x family members = the product of the two denominators = 3*7 = 21.
Number who chose to have a hamburger = (1/3)(21) = 7.
Since 1/7 of the hamburger-eaters also had a hot dog, the number who had both = (1/7)(7) = 1.
Thus, the number who did NOT have both = (total members) - (number who had ...”

“I believe that the prompt above is outdated; regardless, it does not list a correct answer. The prompt below appears in Manhattan''s 5 lb. Book of GRE Practice Problems:
Case 1: 57XXX
Number of options for the hundreds digit = 10. (Any digit 0-9.)
Number of options for the tens digit = 10. ...”

“Target question: When n is divided by 19, what is the remainder?
Statement 1: n-17 is a multiple of 19
----ASIDE-------------------------------------------
If N is a multiple of d, then we can write N = dk (for some integer k)
For example, if N is a multiple of 5, then we can write N = 5k ...”

“points A, B, C, D, and E represent the five teams in a certain league in which each team must play each of the other teams exactly once.
Let''s first determine the total number of games that will be played.
There are 5 teams, so each team will play 4 games (since a team can''t play itself)
So, ...”

“Since the N cookies can be divided evenly among 24 or 18 students, N must be divisible by both 24 and 18.
One way to determine the LCM of two integers is to take multiples of the LARGER integer until we get a multiple of the SMALLER integer.
Multiples of 24:
24, 48, 72...
We can stop here: the ...”

“Statement 1:
If NONE of the members sells at least 2 tickets, then the 3 members sell a maximum of 1 ticket each, yielding the following total:
1+1+1 = 3.
Too small, since the total number of tickets sold = 6.
Implication:
For a total of 6 tickets to be sold, one of the 3 members must sell at ...”

“This is a weighted average/mixture problem.
Average score for the first N games = 35.
Score for the final game = 28.
Average score for the MIXTURE of all the games = 34.
Step-by-step instruction for the alligation approach:
Step 1: Plot the 3 averages on a number line, with the average ...”

“ Given: In four years, Andy will be twice as old as Betsy.
Let A = Andy''s PRESENT age
Let B = Betsy''s PRESENT age
So, A+4 = Andy''s age IN 4 YEARS
And so, B+4 = Betsy''s age IN 4 YEARS
If Andy will be twice as old as Betsy IN 4 YEARS, we can write: A+4 = 2(B+4)
Expand: A + 4 = 2B + 8
...”

“ Given: m and n are integers
Target question: Is m + m² - n an even number?
This is a good candidate for rephrasing the target question
Aside: Here’s a video with tips on rephrasing the target question:
We can take the expression, m + m² - n and rewrite it as m(1 + m) - n
Now notice ...”

“The difference between the endpoints of the range = 200-10 = 190.
Since neither endpoint is a multiple of 3, we can count the multiples of 3 simply by dividing 3 into 190:
190/3 ≈ 63.
The correct answer is D.”

“One approach is to ignore the rule and determine the total number of ways to arrange all 5 people, and then subtract the number of arrangements that BREAK the rule. I''m sure someone will post that kind of solution shortly.
Here''s another approach:
Take the task of arranging the 5 men and ...”

“If x and y are different positive integers, which of the following COULD be true:
i) When x is divided by y, the remainder is 2x
ii) When x is divided by 2y, the remainder is y
iii) When (2x + y) is divided by (x + y), the remainder is y
A) i only
B) ii only
C) iii only
D) i & ii ...”

“i) When x is divided by y, the remainder is x
This occurs any time x < y
For example, if x = 5 and y = 7, then statement i becomes: When 5 is divided by 7, the remainder is 5
So true!
Scan the answer choices....eliminate D
-----------------------------------------------------
ii) When 2x ...”

“Let d=100.
After the value of the stock increases by 40% to $140, the share received by each of the 20 friends = 140/20 = 7.
The correct answer must yield a value of 7 when d=100.
Only E works:
(7d)/100 = (7*100)/100 = 7.
The correct answer is E.”

“Another approach:
The question is really asking, "In how many different ways can we create 2-team pairings from 16 teams?"
Since the order of the selections does not matter (i.e., selecting teams A and B to play, is the same as selecting teams B and A to play), we can use combinations.
...”

“There are 16 teams. If we ask each team, "How many teams did you play?" we''ll find that each team played 15 teams, which gives us a total of 240 games (since 16 x 15 = 240).
From here we need to recognize that each game has been COUNTED TWICE.
For example, if Team A and Team B play ...”

“We can PLUG IN THE ANSWERS and use alligation.
To learn about alligation, check my posts here:
https://www.beatthegmat.com/statistics-average-question-t288903.html
https://www.beatthegmat.com/alligation-method-t293459.html
https://www.beatthegmat.com/mean-t266852.html
Let O = the original ...”

“Statement 1:
Time and rate have a RECIPROCAL relationship.
Since X and Y together take 2/3 as long as Y alone, X and Y together work 3/2 as fast as Y alone.
Thus, if Y''s rate alone = 2 units per hour, then X and Y''s combined rate = (3/2)(2) = 3 units per hour, implying that X''s rate alone = ...”

“To form a triangle, we must select 3 points such that at most 2 are collinear.
Case 1: Select 3 points not on the x-axis
From the 4 points not on the x-axis, the number of ways to choose 3 = 4C3 = (4*3*2)/(3*2*1) = 4.
Case 2: Select 1 point on the x-axis and two points not on the x-axis
From ...”

“ Given: a/(b - c) = 1
Target question: What is the value of (b-c)/b ?
This is a good candidate for rephrasing the target question.
Aside: Here’s a video with tips on rephrasing the target question:
If a/(b - c) = 1, then we know that a = b - c
The target question asks "What is the ...”

“If x and y are different positive integers, which of the following COULD be true:
i) When x is divided by y, the remainder is x
ii) When 2x is divided by y, the remainder is x
iii) When x+y is divided by x , the remainder is x-y
A) i only
B) i & ii only
C) i & iii only
D) ii & ...”

“Number of options for president = 8. (Any of the 8 candidates.)
Number of options for treasurer = 7. (Any of the 7 remaining candidates.)
From the 6 remaining people, the number of ways to choose 3 to serve as warrant officers = 6C3 = (6*5*4)/(3*2*1) = 20.
To combine these options, we multiply:
...”

“For the set of pencils to be shared evenly, the total number of pencils must be divisible by the total number of children.
Thus, the correct answer must be divisible by 2, 3, 4, 5 and 6.
Of the five answer choices, only E is divisible by 2, 3, 4, 5 and 6.
The correct answer is E.”

“The main idea here is that all 3 travelers travel for the SAME amount of time.
So, let''s first determine the time it takes for Joe and Sue to meet.
This is a shrinking gap question.
To determine the rate at which the gap shrinks, we''ll ADD the travel speeds of Joe and Sue
That is, ...”

“Target question: What is the value of y?
Given: x and y are positive
Statement 1: xy is the square of an integer.
There are several values of x and y that satisfy this condition. Here are two:
Case a: x = 1 and y = 1. Here xy = (1)(1) = 1, which is the square of an integer. In this ...”

“At noon, Joe and a hummingbird leave point A and travel towards point B, which is 12 miles away. At the same time Sue leaves point B and travels toward point A. Joe, Sue and the hummingbird travel at constant speeds of 3 miles per hour, 4 miles per hour, and 84 miles per hour respectively. When the ...”

“Target question: What is the probability that Harvey will be paired with Jessica?
Given: There are n students in a class. Of them, k boys and k girls (including Harvey and Jessica) are selected for a dance performance in which students will dance in pairs of one boy and one girl.
...”

“Target question: What is the value of n?
Given: When the positive integer n is divided by 25, the remainder is 13.
-------ASIDE-------------
When it comes to remainders, we have a nice rule that says:
If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, ...”

“As with many probability questions, we can solve this by using counting techniques or rules of probability.
Let''s use rules of probability
P(balloon animal is an elephant) = 1/4
P(balloon animal is purple) = 1/5
P(clown makes a purple elephant balloon animal) = P(balloon animal is ...”